\documentclass[12pt]{standalone}

\usepackage{amsmath,amssymb}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}[x=2cm,y=2cm,every node/.style={draw}]

\node (000) at (0,0) {$\varnothing$};
\node (001) at (-1.5,1) {$\{0\}$};
\node (010) at (0,1) {$\{1\}$};
\node (100) at (1.5,1) {$\{2\}$};
\node (011) at (-1.5,2) {$\{0,1\}$};
\node (101) at (0,2) {$\{0,2\}$};
\node (110) at (1.5,2) {$\{1,2\}$};
\node (111) at (0,3) {$\{0,1,2\}$};

\draw[->,thick] (000) -- (001);
\draw[->,thick] (001) -- (011);
\draw[->,thick] (011) -- (111);
\draw[->,thick] (000) -- (100);
\draw[->,thick] (100) -- (110);
\draw[->,thick] (110) -- (111);
\draw[->,thick] (000) -- (010);
\draw[->,thick] (010) -- (011);
\draw[->,thick] (010) -- (110);
\draw[->,thick] (001) -- (101);
\draw[->,thick] (100) -- (101);
\draw[->,thick] (101) -- (111);

\end{tikzpicture}
\end{document}
